Sine 30 degrees is one of the most important trigonometric values. It has many applications in mathematics and physics.

This article will discuss what sine 30 degrees is, a geometrical proof of it’s value and how to find equivalent trig values.

We will also show a video demonstration of some example problems with solutions.

## What is the exact value of sin 30 degrees?

The value of sine 30 degrees is 1/2.

As a decimal you could say sin 30°=0.5

In radians, sin 30° = sin (**π**/6) = 1/2

## Derivation of sine 30 degrees (Geometrical method)

This can be proved using geometry. Draw an equilateral triangle of length 2cm. Each angle is 60°.

Construct a perpendicular bisector from one side to the opposite angle.

This bisects the 60° angle. So a 30° angle is formed along with a side of 1cm.

Since the trigonometric ratio is sine is equal to the opposite side/ hypotenuse.

Label the 1cm side opposite to the 30° angle as opposite and the 2cm side opposite the right angle as hypotenuse as follows:

So we can see sin 30 = 1/2

This is the exact value of sine 30 degrees.

## Trigonometric Ratios

There are 6 trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec) and cotangent (cot).

Out of these, sine, cosine and tangent are the most important. They are defined as:

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

where the opposite side is opposite the angle of interest usally marked as θ or with an actual value.

The adjacent side is next to the angle of interest.

The hypotenuse is opposite the right angle. It is always the longest side in a right-angled triangle.

The other three trigonometric ratios can be derived from these.

They are reciprocal ratios:

cosecant (cosec) θ = 1/sin θ

secant (sec) θ = 1/cos θ

cotangent (cot) θ = 1/tan θ

## Trigonometry table

Finding the exact values of the common angles 0°, 30°, 45°, 60° and 90° is important for solving many problems.

This trigonometry table shows the exact values of these angles.

## Complementary trig values

Complementary angles are two angles whose sum is equal to 90°.

Finding equivalent trigonometric values is important for solving various trig problems.

In many problems, we are given only one value of a trigonometric ratio. But we need to find another value which is equal to it.

For example, if we are asked to find cos 60° we can say it equals 1/2 too because 60° + 30° = 90°.

So sin 30° = cos 60° = 1/2

## Sine 30 degrees on the Unit Circle

Sine 30 degrees can be found on the unit circle as it is the y co-ordinate of the point that is 30 degrees from the positive direction of the x axis.

As the y coordinate is 0.5, sin 30° = 0.5

## Why is sine 150 degrees equal to sin 30 degrees?

150° = 180°-30°

So sine 150 degress is equal to sine 30 degrees because 150 degrees is in the second quadrant where sine is positive and the related angle is 30 degrees.

## Equivalent values of sin 30

These are some other values which sine 30 can be written as:

sin 30° = sin (150 + 360n)°, where n is an integer

sin 30° = sin (30 + 360n)°, where n is an integer

sin 30° = 1/ cosec 30°

sin 30° = 1/sec 60°

## Example problems

Question: Is -sin 30° the same as sin (-30)°?

Solution: Yes, sin (-30)° is in the 4th quadrant. It is measured clockwise from 0°. Sine is negative in the 4th qudrant, so sin(-30)° = -sin 30° = 1/2

Question: Find the exact value of sin 210°.

Solution: 210° = (180 + 30)° so this is in the 3rd quadrant and 30° is the related angle. Sine is negative in the 3rd quadrant so:

sin 210° = – sin 30°

= – 1/2

Question: Find the exact value of cosec 330°.

Solution: 330° = (360 – 30)° so this is in the 4th quadrant and 30° is the related angle. Cosec is the reciprocal of sine and sine is negative in the 4th quadrant so cosec is also negative.

cosec 330° = 1/ sin 330°

= 1/-sin 30°

= 1/ – 1/2

= -2

Question: A ship sails 50 nautical miles due north from X to Y, then 25 nautical miles due east from Y to Z. Find θ, the bearing of Z from X, correct to the nearest degree.

Solution: Always start with a diagram.

Then using sin θ = opposite/ hypotenuse,

sin θ = 25/50

∴ sin θ = 1/2

∴ θ = 30°

For more example problems on bearings see this article.

## FAQs

### What is the sin and cos of 30 degrees?

Sin 30 degrees is 1/2 and cos 30 degrees equals **√**3/2.

### How do you find sin 30 degrees without a calculator?

Use the unit circle. sin 30 is the y co-ordinate of the point that is 30 degrees above the x axis. This is 0.5

### How do you do sin 30 on a calculator?

Press ‘sin’ then 30 and =

### What is the sin for 30 degrees?

0.5 = 1/2

## Conclusion

Learning the exact value of sin 30 is 1/2 can be really helpful when trying to solve trigonometry problems. I hope this has helped and if you want to learn more about trignonometry, click here. Stay sharp!